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Here is what I got from Wikipedia regarding Relativistic Doppler Effect:
Relativistic Doppler effect
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In physics, the relativistic Doppler effect is change in the observed wavelength and frequency of light due to the relative motion of source and observer when taking into account the Special Theory of Relativity.
In general, the wavelength λ measured at the source and by the distant observer can be expressed as a ratio:
z+1 = \frac{\lambda_{observed}}{\lambda_{emitted}} = (1-\beta cos{\theta})\gamma
Where:
\beta cos{\theta} = \frac{v_\|}{c} = velocity directed towards the line of sight of a photon (v_\|) divided by the speed of light in proper time (c)
γ = Lorentz factor due to the total velocity of the object (not just in the line of sight)
z is the redshift.
For the classical Doppler effect arrangement when the observer and the source are approaching each other and there is no perpendicular movement, the equation reduces to:
z+1 = \frac{\lambda_{observed}}{\lambda_{emitted}} = \sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}} }
Where:
θ = 0
v is the velocity of an object approaching a photon, in meters per "coordinate" second.
c is the speed of light which equals 299,792,458 meters per "proper" second.
Contents
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* 1 Speed of light is consistent with proper time
o 1.1 The geometry of a light interception
* 2 Etc.
* 3 See also
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Speed of light is consistent with proper time
In the Special Theory of Relativity, space and time are not absolute. Neither is speed, which is something like a rotation in space-time, so that two observers travelling at different velocities have different points of view regarding which sets of points are simultaneous with another point in space and time.
A moving observer will go under velocity time dilation so that for every second that passes by in the coordinate time, the processes that animate the observer undergo fewer cycles for a given coordinate time interval, and thus a proper time is established. No matter which direction the observer is travelling, the speed of a photon of light is always consistent with the proper time of the observer's clock (more accurately the photon's clock if you consider general relativistic time delay effects and/or variable index of refraction, variable magnetic permeability, and variable permittivity). A change of speed by the observer does correspond to a sort of rotation in which the time and space axes are different. An application of the Lorentz transformation shows that the effect is that the frequency, and therefore energy, of the photons is affected (because coordinate time/proper time increases): photons from any approaching source are increased in energy and photons from a receding source are reduced in energy: this is more intuitively satisfying.
It is similar to the change in kinetic energy of an object due to the thing or person throwing it moving relative to the observer. This is the relativistic Doppler effect.
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The geometry of a light interception
The following equation is derived from the Lorentz factor.
\left( \frac{d\tau}{dt} \right) ^2 + \left( \frac{v}{c} \right) ^2 =1
The equation above is the same as the one for a circle.
Where:
\frac{d\tau}{dt} is equivalent to proper seconds per coordinate second, the reciprocal of the Lorentz factor. \frac{d\tau}{dt}=\frac{1}{\gamma}. A large value of this is consistent with opposite of time "dilating", the opposite of "slow-motion" time - the equivalent of fast-paced "space".
dτ is the interval of proper time (i.e. of time-dilated time).
dt is the interval of coordinate time (i.e. of non time-dilated time).
dτ < dt
v < c
After mutiplying both sides by c2 we get the following:
\left( \frac{c d\tau}{dt} \right) ^2 + v^2 =c^2
\bar{c}^2 + v^2 =c^2=constant
\bar{c} is the velocity of the light in meters per coordinate second, relative to the observer which moves at a velocity of v in meters per coordinate second.
c is a constant measuring the speed of light in meters per proper (observed) second.
This equation also has the form of a circle, a shape where each point on the line has a common radius with the center.
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Etc.
In this article the term light is intended to mean any electromagnetic radiation, which is a particle that has no rest mass and which must always travel at the speed of light (when in a vacuum). Measurement of a single photon is always done using proper time, and thus the speed of that photon is always measured to be equal to c, no matter how one moves, because the movement itself influences how the photon progates within the observing equipment as well as the molecular processes that is within it.
The article on electromagnetic radiation also gives the relationship between frequency and energy of a particle.
Here is the link:
Relativistic Doppler Effect
